Witnessing sequence for a nondeterministic automaton

For $w=a_0\cdots a_{n-1}$, a witnessing sequence from $p_0$ to $p_n$ satisfies $p_{i+1}\in \Delta (p_i,a_i)$. Induction on word length shows that $q^{\prime }\in \hat{\Delta }(q,w)$ exactly when such a sequence runs from $q$ to $q^{\prime }$.